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The natural logarithm function, often written as \ln(x)\, is a fundamental mathematical function with a wide range of applications. At its core, a natural logarithm answers the question: "To what power must the base ***e*** be raised to get ***x***?" Here, ***e*** is a mathematical constant approximately equal to 2.71828.

A critical point to remember about natural logarithms is that they are only defined for positive values. In other words, the argument of a natural logarithm must always be positive. This implies \( x > 0 \) for the function \( \ln(x) \) to exist. In scenarios involving more complex expressions, such as \( \ln \left( \frac{x}{x+1} \right) \), the entire expression \( \frac{x}{x+1} \) should be greater than zero to ensure the logarithm is defined.

Natural logarithms have properties that can simplify various algebraic manipulations such as:

• \( \ln(ab) = \ln(a) + \ln(b) \) - the logarithm of a product.
• \( \ln \left( \frac{a}{b} \right) = \ln(a) - \ln(b) \) - the logarithm of a quotient.
• \( \ln(a^b) = b \cdot \ln(a) \) - the logarithm of a power.

These properties become extremely useful when solving equations or inequalities involving logarithmic functions.

Inequalities are mathematical expressions involving relations like greater than (>), less than (<), greater than or equal to (≥), and less than or equal to (≤). Solving an inequality can sometimes be like solving equations, but with some important differences.

When dealing with inequalities, particularly those involving fractions or rational expressions like \( \frac{x}{x+1} \), it's essential to consider the critical points that make either the numerator or denominator zero. Identifying these points helps break down the inequality into intervals.

For \( \frac{x}{x+1} > 0 \), you would:

• Identify the zeros: \( x = 0 \) (numerator) and the undefined point \( x = -1 \) (denominator).
• Analyze the signs over the intervals created by these critical points: \((-\infty, -1)\), \((-1, 0)\), and \((0, \infty)\). This involves checking if the entire expression is positive or negative within each interval.

Remember, multiplying or dividing by a negative number reverses the inequality sign. Also, always express solutions as intervals to showcase where the inequality holds true.

The domain of a function is a set of all possible input values (or x-values) for which the function is defined. Understanding the domain is crucial as it delineates where a function can safely be evaluated without running into undefined operations like division by zero or taking the logarithm of a non-positive value.

For the function \( f(x) = \ln \frac{x}{x+1} \), the domain is determined by ensuring the expression inside the logarithm stays positive and avoids division by zero. Thus, you first find where \( \frac{x}{x+1} > 0 \), carefully checking critical points like \( x = -1 \) where the expression becomes undefined.

The intervals \((-\infty, -1) \cup (0, \infty)\) define the domain in this case. This indicates:

• For \( x \in (-\infty, -1) \), both the numerator and the denominator are negative, keeping \( \frac{x}{x+1} \) positive.
• For \( x \in (0, \infty) \), both the numerator and the denominator are positive, also resulting in a positive quotient.

Identifying an expression’s domain helps understand the behavior and limitations of the function, ensuring valid evaluations across specified intervals.